| Step | Hyp | Ref
| Expression |
| 1 | | etransclem48.q |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ((Poly‘ℤ) ∖
{0𝑝})) |
| 2 | 1 | eldifad 3963 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈
(Poly‘ℤ)) |
| 3 | | 0zd 12625 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈
ℤ) |
| 4 | | etransclem48.a |
. . . . . . . . . 10
⊢ 𝐴 = (coeff‘𝑄) |
| 5 | 4 | coef2 26270 |
. . . . . . . . 9
⊢ ((𝑄 ∈ (Poly‘ℤ)
∧ 0 ∈ ℤ) → 𝐴:ℕ0⟶ℤ) |
| 6 | 2, 3, 5 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → 𝐴:ℕ0⟶ℤ) |
| 7 | | 0nn0 12541 |
. . . . . . . . 9
⊢ 0 ∈
ℕ0 |
| 8 | 7 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℕ0) |
| 9 | 6, 8 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (𝜑 → (𝐴‘0) ∈ ℤ) |
| 10 | | zabscl 15352 |
. . . . . . 7
⊢ ((𝐴‘0) ∈ ℤ →
(abs‘(𝐴‘0))
∈ ℤ) |
| 11 | 9, 10 | syl 17 |
. . . . . 6
⊢ (𝜑 → (abs‘(𝐴‘0)) ∈
ℤ) |
| 12 | | etransclem48.m |
. . . . . . . . 9
⊢ 𝑀 = (deg‘𝑄) |
| 13 | | dgrcl 26272 |
. . . . . . . . . 10
⊢ (𝑄 ∈ (Poly‘ℤ)
→ (deg‘𝑄) ∈
ℕ0) |
| 14 | 2, 13 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (deg‘𝑄) ∈
ℕ0) |
| 15 | 12, 14 | eqeltrid 2845 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 16 | 15 | faccld 14323 |
. . . . . . 7
⊢ (𝜑 → (!‘𝑀) ∈ ℕ) |
| 17 | 16 | nnzd 12640 |
. . . . . 6
⊢ (𝜑 → (!‘𝑀) ∈ ℤ) |
| 18 | | ssrab2 4080 |
. . . . . . . 8
⊢ {𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ⊆
ℕ0 |
| 19 | | nn0ssz 12636 |
. . . . . . . 8
⊢
ℕ0 ⊆ ℤ |
| 20 | 18, 19 | sstri 3993 |
. . . . . . 7
⊢ {𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ⊆ ℤ |
| 21 | | etransclem48.i |
. . . . . . . 8
⊢ 𝐼 = inf({𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}, ℝ, < ) |
| 22 | | nn0uz 12920 |
. . . . . . . . . 10
⊢
ℕ0 = (ℤ≥‘0) |
| 23 | 18, 22 | sseqtri 4032 |
. . . . . . . . 9
⊢ {𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ⊆
(ℤ≥‘0) |
| 24 | | 1rp 13038 |
. . . . . . . . . . 11
⊢ 1 ∈
ℝ+ |
| 25 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝜑 |
| 26 | | nfmpt1 5250 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ 𝐶) |
| 27 | | nfmpt1 5250 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) |
| 28 | | etransclem48.s |
. . . . . . . . . . . . . . 15
⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 29 | | nfmpt1 5250 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑛(𝑛 ∈ ℕ0 ↦ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 30 | 28, 29 | nfcxfr 2903 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑆 |
| 31 | | nn0ex 12532 |
. . . . . . . . . . . . . . . . 17
⊢
ℕ0 ∈ V |
| 32 | 31 | mptex 7243 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ 𝐶) ∈
V |
| 33 | 32 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ 𝐶) ∈ V) |
| 34 | | etransclem48.c |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) |
| 35 | | fzfid 14014 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (0...𝑀) ∈ Fin) |
| 36 | 6 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝐴:ℕ0⟶ℤ) |
| 37 | | elfznn0 13660 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0) |
| 38 | 37 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℕ0) |
| 39 | 36, 38 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐴‘𝑗) ∈ ℤ) |
| 40 | 39 | zcnd 12723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝐴‘𝑗) ∈ ℂ) |
| 41 | | ere 16125 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ e ∈
ℝ |
| 42 | 41 | recni 11275 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ e ∈
ℂ |
| 43 | 42 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → e ∈ ℂ) |
| 44 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ) |
| 45 | 44 | zcnd 12723 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ) |
| 46 | 45 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑗 ∈ ℂ) |
| 47 | 43, 46 | cxpcld 26750 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (e↑𝑐𝑗) ∈
ℂ) |
| 48 | 40, 47 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝐴‘𝑗) · (e↑𝑐𝑗)) ∈
ℂ) |
| 49 | 48 | abscld 15475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) ∈
ℝ) |
| 50 | 49 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) ∈
ℂ) |
| 51 | 15 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 52 | | peano2nn0 12566 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℕ0
→ (𝑀 + 1) ∈
ℕ0) |
| 53 | 15, 52 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑀 + 1) ∈
ℕ0) |
| 54 | 51, 53 | expcld 14186 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑀↑(𝑀 + 1)) ∈ ℂ) |
| 55 | 51, 54 | mulcld 11281 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑀 · (𝑀↑(𝑀 + 1))) ∈ ℂ) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 · (𝑀↑(𝑀 + 1))) ∈ ℂ) |
| 57 | 50, 56 | mulcld 11281 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) ∈ ℂ) |
| 58 | 35, 57 | fsumcl 15769 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) ∈ ℂ) |
| 59 | 34, 58 | eqeltrid 2845 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐶 ∈ ℂ) |
| 60 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝑛 ∈ ℕ0
↦ 𝐶) = (𝑛 ∈ ℕ0
↦ 𝐶)) |
| 61 | | eqidd 2738 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ ℕ0) ∧ 𝑛 = 𝑖) → 𝐶 = 𝐶) |
| 62 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈
ℕ0) |
| 63 | 59 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐶 ∈
ℂ) |
| 64 | 60, 61, 62, 63 | fvmptd 7023 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ 𝐶)‘𝑖) = 𝐶) |
| 65 | 22, 3, 33, 59, 64 | climconst 15579 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ 𝐶) ⇝ 𝐶) |
| 66 | 31 | mptex 7243 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ (𝐶 ·
(((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) ∈ V |
| 67 | 28, 66 | eqeltri 2837 |
. . . . . . . . . . . . . . 15
⊢ 𝑆 ∈ V |
| 68 | 67 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ∈ V) |
| 69 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) |
| 70 | 69 | expfac 45672 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀↑(𝑀 + 1)) ∈ ℂ → (𝑛 ∈ ℕ0
↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) ⇝ 0) |
| 71 | 54, 70 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) ⇝ 0) |
| 72 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℕ0) |
| 73 | 59 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈
ℂ) |
| 74 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ0
↦ 𝐶) = (𝑛 ∈ ℕ0
↦ 𝐶) |
| 75 | 74 | fvmpt2 7027 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ ℕ0
∧ 𝐶 ∈ ℂ)
→ ((𝑛 ∈
ℕ0 ↦ 𝐶)‘𝑛) = 𝐶) |
| 76 | 72, 73, 75 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ 𝐶)‘𝑛) = 𝐶) |
| 77 | 76, 73 | eqeltrd 2841 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ 𝐶)‘𝑛) ∈
ℂ) |
| 78 | | ovex 7464 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)) ∈ V |
| 79 | 69 | fvmpt2 7027 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ0
∧ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)) ∈ V) → ((𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛) = (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) |
| 80 | 78, 79 | mpan2 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛) = (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) |
| 81 | 80 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛) = (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) |
| 82 | 54 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑀↑(𝑀 + 1)) ∈ ℂ) |
| 83 | 82, 72 | expcld 14186 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑀↑(𝑀 + 1))↑𝑛) ∈ ℂ) |
| 84 | 72 | faccld 14323 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(!‘𝑛) ∈
ℕ) |
| 85 | 84 | nncnd 12282 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(!‘𝑛) ∈
ℂ) |
| 86 | 84 | nnne0d 12316 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(!‘𝑛) ≠
0) |
| 87 | 83, 85, 86 | divcld 12043 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)) ∈ ℂ) |
| 88 | 81, 87 | eqeltrd 2841 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛) ∈ ℂ) |
| 89 | | ovex 7464 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) ∈ V |
| 90 | 28 | fvmpt2 7027 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ0
∧ (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) ∈ V) → (𝑆‘𝑛) = (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 91 | 89, 90 | mpan2 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
→ (𝑆‘𝑛) = (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 92 | 91 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑆‘𝑛) = (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 93 | 76, 81 | oveq12d 7449 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (((𝑛 ∈ ℕ0
↦ 𝐶)‘𝑛) · ((𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛)) = (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))) |
| 94 | 92, 93 | eqtr4d 2780 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑆‘𝑛) = (((𝑛 ∈ ℕ0 ↦ 𝐶)‘𝑛) · ((𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛))) |
| 95 | 25, 26, 27, 30, 22, 3, 65, 68, 71, 77, 88, 94 | climmulf 45619 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ⇝ (𝐶 · 0)) |
| 96 | 59 | mul01d 11460 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐶 · 0) = 0) |
| 97 | 95, 96 | breqtrd 5169 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ⇝ 0) |
| 98 | | eqidd 2738 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑆‘𝑛) = (𝑆‘𝑛)) |
| 99 | 77, 88 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (((𝑛 ∈ ℕ0
↦ 𝐶)‘𝑛) · ((𝑛 ∈ ℕ0 ↦ (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)))‘𝑛)) ∈ ℂ) |
| 100 | 94, 99 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝑆‘𝑛) ∈ ℂ) |
| 101 | 30, 22, 3, 68, 98, 100 | clim0cf 45669 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑆 ⇝ 0 ↔ ∀𝑒 ∈ ℝ+ ∃𝑖 ∈ ℕ0
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 𝑒)) |
| 102 | 97, 101 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑖 ∈ ℕ0
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 𝑒) |
| 103 | | breq2 5147 |
. . . . . . . . . . . . 13
⊢ (𝑒 = 1 → ((abs‘(𝑆‘𝑛)) < 𝑒 ↔ (abs‘(𝑆‘𝑛)) < 1)) |
| 104 | 103 | rexralbidv 3223 |
. . . . . . . . . . . 12
⊢ (𝑒 = 1 → (∃𝑖 ∈ ℕ0
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 𝑒 ↔ ∃𝑖 ∈ ℕ0 ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1)) |
| 105 | 104 | rspcva 3620 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ+ ∧ ∀𝑒 ∈ ℝ+ ∃𝑖 ∈ ℕ0
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 𝑒) → ∃𝑖 ∈ ℕ0 ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1) |
| 106 | 24, 102, 105 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1) |
| 107 | | rabn0 4389 |
. . . . . . . . . 10
⊢ ({𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ≠ ∅ ↔ ∃𝑖 ∈ ℕ0
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1) |
| 108 | 106, 107 | sylibr 234 |
. . . . . . . . 9
⊢ (𝜑 → {𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ≠ ∅) |
| 109 | | infssuzcl 12974 |
. . . . . . . . 9
⊢ (({𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ⊆
(ℤ≥‘0) ∧ {𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ≠ ∅) → inf({𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}, ℝ, < ) ∈ {𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}) |
| 110 | 23, 108, 109 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → inf({𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}, ℝ, < ) ∈ {𝑖 ∈ ℕ0
∣ ∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}) |
| 111 | 21, 110 | eqeltrid 2845 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ {𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}) |
| 112 | 20, 111 | sselid 3981 |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ ℤ) |
| 113 | | tpssi 4838 |
. . . . . 6
⊢
(((abs‘(𝐴‘0)) ∈ ℤ ∧
(!‘𝑀) ∈ ℤ
∧ 𝐼 ∈ ℤ)
→ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℤ) |
| 114 | 11, 17, 112, 113 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℤ) |
| 115 | | etransclem48.t |
. . . . . 6
⊢ 𝑇 = sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
) |
| 116 | | xrltso 13183 |
. . . . . . . 8
⊢ < Or
ℝ* |
| 117 | 116 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → < Or
ℝ*) |
| 118 | | tpfi 9365 |
. . . . . . . 8
⊢
{(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ∈ Fin |
| 119 | 118 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ∈ Fin) |
| 120 | 11 | tpnzd 4780 |
. . . . . . 7
⊢ (𝜑 → {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ≠ ∅) |
| 121 | | zssre 12620 |
. . . . . . . . 9
⊢ ℤ
⊆ ℝ |
| 122 | | ressxr 11305 |
. . . . . . . . 9
⊢ ℝ
⊆ ℝ* |
| 123 | 121, 122 | sstri 3993 |
. . . . . . . 8
⊢ ℤ
⊆ ℝ* |
| 124 | 114, 123 | sstrdi 3996 |
. . . . . . 7
⊢ (𝜑 → {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆
ℝ*) |
| 125 | | fisupcl 9509 |
. . . . . . 7
⊢ (( <
Or ℝ* ∧ ({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ∈ Fin ∧ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ≠ ∅ ∧ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℝ*)) →
sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, < ) ∈
{(abs‘(𝐴‘0)),
(!‘𝑀), 𝐼}) |
| 126 | 117, 119,
120, 124, 125 | syl13anc 1374 |
. . . . . 6
⊢ (𝜑 → sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, < ) ∈
{(abs‘(𝐴‘0)),
(!‘𝑀), 𝐼}) |
| 127 | 115, 126 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}) |
| 128 | 114, 127 | sseldd 3984 |
. . . 4
⊢ (𝜑 → 𝑇 ∈ ℤ) |
| 129 | | 0red 11264 |
. . . . 5
⊢ (𝜑 → 0 ∈
ℝ) |
| 130 | 16 | nnred 12281 |
. . . . 5
⊢ (𝜑 → (!‘𝑀) ∈ ℝ) |
| 131 | 128 | zred 12722 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 132 | 16 | nngt0d 12315 |
. . . . 5
⊢ (𝜑 → 0 < (!‘𝑀)) |
| 133 | | fvex 6919 |
. . . . . . . 8
⊢
(!‘𝑀) ∈
V |
| 134 | 133 | tpid2 4770 |
. . . . . . 7
⊢
(!‘𝑀) ∈
{(abs‘(𝐴‘0)),
(!‘𝑀), 𝐼} |
| 135 | | supxrub 13366 |
. . . . . . 7
⊢
(({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℝ* ∧
(!‘𝑀) ∈
{(abs‘(𝐴‘0)),
(!‘𝑀), 𝐼}) → (!‘𝑀) ≤ sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 136 | 124, 134,
135 | sylancl 586 |
. . . . . 6
⊢ (𝜑 → (!‘𝑀) ≤ sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 137 | 136, 115 | breqtrrdi 5185 |
. . . . 5
⊢ (𝜑 → (!‘𝑀) ≤ 𝑇) |
| 138 | 129, 130,
131, 132, 137 | ltletrd 11421 |
. . . 4
⊢ (𝜑 → 0 < 𝑇) |
| 139 | | elnnz 12623 |
. . . 4
⊢ (𝑇 ∈ ℕ ↔ (𝑇 ∈ ℤ ∧ 0 <
𝑇)) |
| 140 | 128, 138,
139 | sylanbrc 583 |
. . 3
⊢ (𝜑 → 𝑇 ∈ ℕ) |
| 141 | | prmunb 16952 |
. . 3
⊢ (𝑇 ∈ ℕ →
∃𝑝 ∈ ℙ
𝑇 < 𝑝) |
| 142 | 140, 141 | syl 17 |
. 2
⊢ (𝜑 → ∃𝑝 ∈ ℙ 𝑇 < 𝑝) |
| 143 | 1 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑄 ∈ ((Poly‘ℤ) ∖
{0𝑝})) |
| 144 | | etransclem48.qe0 |
. . . . 5
⊢ (𝜑 → (𝑄‘e) = 0) |
| 145 | 144 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝑄‘e) = 0) |
| 146 | | etransclem48.a0 |
. . . . 5
⊢ (𝜑 → (𝐴‘0) ≠ 0) |
| 147 | 146 | 3ad2ant1 1134 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝐴‘0) ≠ 0) |
| 148 | | simp2 1138 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑝 ∈ ℙ) |
| 149 | 9 | zcnd 12723 |
. . . . . . 7
⊢ (𝜑 → (𝐴‘0) ∈ ℂ) |
| 150 | 149 | 3ad2ant1 1134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝐴‘0) ∈ ℂ) |
| 151 | 150 | abscld 15475 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (abs‘(𝐴‘0)) ∈ ℝ) |
| 152 | 131 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑇 ∈ ℝ) |
| 153 | | prmz 16712 |
. . . . . . 7
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℤ) |
| 154 | 153 | zred 12722 |
. . . . . 6
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℝ) |
| 155 | 154 | 3ad2ant2 1135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑝 ∈ ℝ) |
| 156 | 124 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆
ℝ*) |
| 157 | | fvex 6919 |
. . . . . . . . 9
⊢
(abs‘(𝐴‘0)) ∈ V |
| 158 | 157 | tpid1 4768 |
. . . . . . . 8
⊢
(abs‘(𝐴‘0)) ∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} |
| 159 | | supxrub 13366 |
. . . . . . . 8
⊢
(({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℝ* ∧
(abs‘(𝐴‘0))
∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}) → (abs‘(𝐴‘0)) ≤ sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 160 | 156, 158,
159 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (abs‘(𝐴‘0)) ≤
sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 161 | 160, 115 | breqtrrdi 5185 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (abs‘(𝐴‘0)) ≤ 𝑇) |
| 162 | 161 | 3adant3 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (abs‘(𝐴‘0)) ≤ 𝑇) |
| 163 | | simp3 1139 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑇 < 𝑝) |
| 164 | 151, 152,
155, 162, 163 | lelttrd 11419 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (abs‘(𝐴‘0)) < 𝑝) |
| 165 | 130 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (!‘𝑀) ∈ ℝ) |
| 166 | 137 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (!‘𝑀) ≤ 𝑇) |
| 167 | 165, 152,
155, 166, 163 | lelttrd 11419 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (!‘𝑀) < 𝑝) |
| 168 | 34 | a1i 11 |
. . . . . . . . 9
⊢ (𝑛 = (𝑝 − 1) → 𝐶 = Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1))))) |
| 169 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑝 − 1) → ((𝑀↑(𝑀 + 1))↑𝑛) = ((𝑀↑(𝑀 + 1))↑(𝑝 − 1))) |
| 170 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑝 − 1) → (!‘𝑛) = (!‘(𝑝 − 1))) |
| 171 | 169, 170 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑛 = (𝑝 − 1) → (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛)) = (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) |
| 172 | 168, 171 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑛 = (𝑝 − 1) → (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) = (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))))) |
| 173 | | prmnn 16711 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℕ) |
| 174 | | nnm1nn0 12567 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ℕ → (𝑝 − 1) ∈
ℕ0) |
| 175 | 173, 174 | syl 17 |
. . . . . . . . 9
⊢ (𝑝 ∈ ℙ → (𝑝 − 1) ∈
ℕ0) |
| 176 | 175 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑝 − 1) ∈
ℕ0) |
| 177 | 58 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) ∈ ℂ) |
| 178 | 54 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑀↑(𝑀 + 1)) ∈ ℂ) |
| 179 | 178, 176 | expcld 14186 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → ((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) ∈ ℂ) |
| 180 | 175 | faccld 14323 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ ℙ →
(!‘(𝑝 − 1))
∈ ℕ) |
| 181 | 180 | nncnd 12282 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ ℙ →
(!‘(𝑝 − 1))
∈ ℂ) |
| 182 | 181 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (!‘(𝑝 − 1)) ∈
ℂ) |
| 183 | 180 | nnne0d 12316 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ ℙ →
(!‘(𝑝 − 1))
≠ 0) |
| 184 | 183 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (!‘(𝑝 − 1)) ≠
0) |
| 185 | 179, 182,
184 | divcld 12043 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))) ∈
ℂ) |
| 186 | 177, 185 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) ∈
ℂ) |
| 187 | 28, 172, 176, 186 | fvmptd3 7039 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑆‘(𝑝 − 1)) = (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))))) |
| 188 | 187 | eqcomd 2743 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) = (𝑆‘(𝑝 − 1))) |
| 189 | 188 | 3adant3 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) = (𝑆‘(𝑝 − 1))) |
| 190 | 112 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 ∈ ℤ) |
| 191 | | 1zzd 12648 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ ℙ → 1 ∈
ℤ) |
| 192 | 153, 191 | zsubcld 12727 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ℙ → (𝑝 − 1) ∈
ℤ) |
| 193 | 192 | 3ad2ant2 1135 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝑝 − 1) ∈ ℤ) |
| 194 | 190 | zred 12722 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 ∈ ℝ) |
| 195 | | tpid3g 4772 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ ℤ → 𝐼 ∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}) |
| 196 | 112, 195 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}) |
| 197 | | supxrub 13366 |
. . . . . . . . . . . . . 14
⊢
(({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼} ⊆ ℝ* ∧ 𝐼 ∈ {(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}) → 𝐼 ≤ sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 198 | 124, 196,
197 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ≤ sup({(abs‘(𝐴‘0)), (!‘𝑀), 𝐼}, ℝ*, <
)) |
| 199 | 198, 115 | breqtrrdi 5185 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼 ≤ 𝑇) |
| 200 | 199 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 ≤ 𝑇) |
| 201 | 194, 152,
155, 200, 163 | lelttrd 11419 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 < 𝑝) |
| 202 | 153 | 3ad2ant2 1135 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝑝 ∈ ℤ) |
| 203 | | zltlem1 12670 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ ℤ ∧ 𝑝 ∈ ℤ) → (𝐼 < 𝑝 ↔ 𝐼 ≤ (𝑝 − 1))) |
| 204 | 190, 202,
203 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝐼 < 𝑝 ↔ 𝐼 ≤ (𝑝 − 1))) |
| 205 | 201, 204 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 ≤ (𝑝 − 1)) |
| 206 | | eluz2 12884 |
. . . . . . . . 9
⊢ ((𝑝 − 1) ∈
(ℤ≥‘𝐼) ↔ (𝐼 ∈ ℤ ∧ (𝑝 − 1) ∈ ℤ ∧ 𝐼 ≤ (𝑝 − 1))) |
| 207 | 190, 193,
205, 206 | syl3anbrc 1344 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝑝 − 1) ∈
(ℤ≥‘𝐼)) |
| 208 | 111 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 𝐼 ∈ {𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1}) |
| 209 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐼 → (ℤ≥‘𝑖) =
(ℤ≥‘𝐼)) |
| 210 | 209 | raleqdv 3326 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐼 → (∀𝑛 ∈ (ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1 ↔ ∀𝑛 ∈ (ℤ≥‘𝐼)(abs‘(𝑆‘𝑛)) < 1)) |
| 211 | 210 | elrab 3692 |
. . . . . . . . . 10
⊢ (𝐼 ∈ {𝑖 ∈ ℕ0 ∣
∀𝑛 ∈
(ℤ≥‘𝑖)(abs‘(𝑆‘𝑛)) < 1} ↔ (𝐼 ∈ ℕ0 ∧
∀𝑛 ∈
(ℤ≥‘𝐼)(abs‘(𝑆‘𝑛)) < 1)) |
| 212 | 208, 211 | sylib 218 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝐼 ∈ ℕ0 ∧
∀𝑛 ∈
(ℤ≥‘𝐼)(abs‘(𝑆‘𝑛)) < 1)) |
| 213 | 212 | simprd 495 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → ∀𝑛 ∈ (ℤ≥‘𝐼)(abs‘(𝑆‘𝑛)) < 1) |
| 214 | | nfcv 2905 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛abs |
| 215 | | nfcv 2905 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛(𝑝 − 1) |
| 216 | 30, 215 | nffv 6916 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛(𝑆‘(𝑝 − 1)) |
| 217 | 214, 216 | nffv 6916 |
. . . . . . . . . 10
⊢
Ⅎ𝑛(abs‘(𝑆‘(𝑝 − 1))) |
| 218 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑛
< |
| 219 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑛1 |
| 220 | 217, 218,
219 | nfbr 5190 |
. . . . . . . . 9
⊢
Ⅎ𝑛(abs‘(𝑆‘(𝑝 − 1))) < 1 |
| 221 | | 2fveq3 6911 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑝 − 1) → (abs‘(𝑆‘𝑛)) = (abs‘(𝑆‘(𝑝 − 1)))) |
| 222 | 221 | breq1d 5153 |
. . . . . . . . 9
⊢ (𝑛 = (𝑝 − 1) → ((abs‘(𝑆‘𝑛)) < 1 ↔ (abs‘(𝑆‘(𝑝 − 1))) < 1)) |
| 223 | 220, 222 | rspc 3610 |
. . . . . . . 8
⊢ ((𝑝 − 1) ∈
(ℤ≥‘𝐼) → (∀𝑛 ∈ (ℤ≥‘𝐼)(abs‘(𝑆‘𝑛)) < 1 → (abs‘(𝑆‘(𝑝 − 1))) < 1)) |
| 224 | 207, 213,
223 | sylc 65 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (abs‘(𝑆‘(𝑝 − 1))) < 1) |
| 225 | 171 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑝 − 1) → (𝐶 · (((𝑀↑(𝑀 + 1))↑𝑛) / (!‘𝑛))) = (𝐶 · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))))) |
| 226 | | ovexd 7466 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝐶 · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) ∈ V) |
| 227 | 28, 225, 176, 226 | fvmptd3 7039 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑆‘(𝑝 − 1)) = (𝐶 · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))))) |
| 228 | 15 | nn0red 12588 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 229 | 228, 53 | reexpcld 14203 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀↑(𝑀 + 1)) ∈ ℝ) |
| 230 | 228, 229 | remulcld 11291 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 · (𝑀↑(𝑀 + 1))) ∈ ℝ) |
| 231 | 230 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 · (𝑀↑(𝑀 + 1))) ∈ ℝ) |
| 232 | 49, 231 | remulcld 11291 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) ∈ ℝ) |
| 233 | 35, 232 | fsumrecl 15770 |
. . . . . . . . . . . . 13
⊢ (𝜑 → Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) ∈ ℝ) |
| 234 | 34, 233 | eqeltrid 2845 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 235 | 234 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → 𝐶 ∈ ℝ) |
| 236 | 229 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑀↑(𝑀 + 1)) ∈ ℝ) |
| 237 | 236, 176 | reexpcld 14203 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → ((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) ∈ ℝ) |
| 238 | 180 | nnred 12281 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ℙ →
(!‘(𝑝 − 1))
∈ ℝ) |
| 239 | 238 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (!‘(𝑝 − 1)) ∈
ℝ) |
| 240 | 237, 239,
184 | redivcld 12095 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1))) ∈
ℝ) |
| 241 | 235, 240 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝐶 · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) ∈
ℝ) |
| 242 | 227, 241 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ) → (𝑆‘(𝑝 − 1)) ∈ ℝ) |
| 243 | 242 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝑆‘(𝑝 − 1)) ∈ ℝ) |
| 244 | | 1red 11262 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → 1 ∈ ℝ) |
| 245 | 243, 244 | absltd 15468 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → ((abs‘(𝑆‘(𝑝 − 1))) < 1 ↔ (-1 < (𝑆‘(𝑝 − 1)) ∧ (𝑆‘(𝑝 − 1)) < 1))) |
| 246 | 224, 245 | mpbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (-1 < (𝑆‘(𝑝 − 1)) ∧ (𝑆‘(𝑝 − 1)) < 1)) |
| 247 | 246 | simprd 495 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (𝑆‘(𝑝 − 1)) < 1) |
| 248 | 189, 247 | eqbrtrd 5165 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → (Σ𝑗 ∈ (0...𝑀)((abs‘((𝐴‘𝑗) · (e↑𝑐𝑗))) · (𝑀 · (𝑀↑(𝑀 + 1)))) · (((𝑀↑(𝑀 + 1))↑(𝑝 − 1)) / (!‘(𝑝 − 1)))) < 1) |
| 249 | | etransclem6 46255 |
. . . 4
⊢ (𝑦 ∈ ℝ ↦ ((𝑦↑(𝑝 − 1)) · ∏𝑧 ∈ (1...𝑀)((𝑦 − 𝑧)↑𝑝))) = (𝑥 ∈ ℝ ↦ ((𝑥↑(𝑝 − 1)) · ∏𝑗 ∈ (1...𝑀)((𝑥 − 𝑗)↑𝑝))) |
| 250 | | eqid 2737 |
. . . 4
⊢
Σ𝑗 ∈
(0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ ((𝑦↑(𝑝 − 1)) · ∏𝑧 ∈ (1...𝑀)((𝑦 − 𝑧)↑𝑝)))‘𝑥)) d𝑥) = Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ ((𝑦↑(𝑝 − 1)) · ∏𝑧 ∈ (1...𝑀)((𝑦 − 𝑧)↑𝑝)))‘𝑥)) d𝑥) |
| 251 | | eqid 2737 |
. . . 4
⊢
(Σ𝑗 ∈
(0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ ((𝑦↑(𝑝 − 1)) · ∏𝑧 ∈ (1...𝑀)((𝑦 − 𝑧)↑𝑝)))‘𝑥)) d𝑥) / (!‘(𝑝 − 1))) = (Σ𝑗 ∈ (0...𝑀)(((𝐴‘𝑗) · (e↑𝑐𝑗)) · ∫(0(,)𝑗)((e↑𝑐-𝑥) · ((𝑦 ∈ ℝ ↦ ((𝑦↑(𝑝 − 1)) · ∏𝑧 ∈ (1...𝑀)((𝑦 − 𝑧)↑𝑝)))‘𝑥)) d𝑥) / (!‘(𝑝 − 1))) |
| 252 | 143, 145,
4, 147, 12, 148, 164, 167, 248, 249, 250, 251 | etransclem47 46296 |
. . 3
⊢ ((𝜑 ∧ 𝑝 ∈ ℙ ∧ 𝑇 < 𝑝) → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1)) |
| 253 | 252 | rexlimdv3a 3159 |
. 2
⊢ (𝜑 → (∃𝑝 ∈ ℙ 𝑇 < 𝑝 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1))) |
| 254 | 142, 253 | mpd 15 |
1
⊢ (𝜑 → ∃𝑘 ∈ ℤ (𝑘 ≠ 0 ∧ (abs‘𝑘) < 1)) |